Problem: The sum of two numbers is $157$, and their difference is $31$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 157}$ ${x-y = 31}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 188 $ $ x = \dfrac{188}{2} $ ${x = 94}$ Now that you know ${x = 94}$ , plug it back into $ {x+y = 157}$ to find $y$ ${(94)}{ + y = 157}$ ${y = 63}$ You can also plug ${x = 94}$ into $ {x-y = 31}$ and get the same answer for $y$ ${(94)}{ - y = 31}$ ${y = 63}$ Therefore, the larger number is $94$, and the smaller number is $63$.